After serving from 1852 to 1854 as a fellow and lecturer at Peterhouse College, Cambridge, England, Tait took a professorship in mathematics at Queen’s College, Belfast, Ireland. There he joined the noted Irish chemist Thomas Andrews in research on the density of ozone and the effect of electric discharges on oxygen and other gases. From 1860 he was a professor of natural philosophy at the University of Edinburgh.

Tait made fundamental contributions to the theory of quaternions, as evident in Elementary Treatise on Quaternions (1867), which went through three editions. Later he wrote Introduction to Quaternions (1873) with Philip Kelland. In collaboration with the English physicist Sir William Thomson (later Lord Kelvin), Tait produced Treatise on Natural Philosophy (1867), which traced the concept of conservation of energy to the work of Sir Isaac Newton. Their efforts were vital to the newly emerging concept of energy and its properties.

After the publication of the Treatise, Tait concentrated on studies of thermoelectricity and thermal conductivity (the capacity for heat flow). His Sketch of the History of Thermodynamics (1868) was highly controversial because of its British bias. His other work includes a pioneering study in the topology of knots (1876–84), an important series of papers on the kinetic theory of gases (1886–92), and classic papers on the trajectory of the golf ball (1890–93). With the Scottish physicist Balfour Stewart, he wrote The Unseen Universe (1867). Such was the public response that they released a sequel, Paradoxical Philosophy (1878).

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